Introduction.—Increased interest in legal matters on the part of actuaries.—Practical nature of legal questions in Institute examinations.—Need for information not found in ordinary text-books.—Why particulars of decided cases are given.—Reporting assurance cases in Journal.—Recent papers dealing with legal subjects.

Nature of the Contract of Life Assurance.—Assurances where a larger amount is payable on maturity than on death.—Are these contracts of life assurance?—Essentials of contract of assurance.—Accident policies with return of premium on death or attainment of fixed age.—Test of a double contract.

Insurable Interest.—Must the interest be a pecuniary one ?— Has an employer an insurable interest in the life of his weekly servant ?—Policy assigned immediately after issue.—Who can plead absence of insurable interest?—Return of premiums when assurance is void.—Agent as expert in assurance law.

The Proposal.—Agent's knowledge as to misstatements in proposal.—Agent as agent of proposer and not of company.—When the agent is the agent of the company.—Conflicting decisions.

The question of the premiums which should be charged to meet the eases of lives proposed for assurance admittedly incurring, or about to incur, risks more hazardous than those of the ordinary assured life, has from time to time engaged the attention of the Institute.

More stress has, perhaps, been laid upon the mortality of persons living in tropical countries, where the conditions of life are very different from those to which we are here accustomed, and statistics, obtained either from official sources or from business experience, have been furnished; whilst the only occupations to which attention has been directed are the Naval, Military, and Mercantile Marine Services, and those connected with the sale of alcohol.

With the view of obtaining, if possible, information respecting the modern practice of Life Assurance in connection with extra risks, I ventured to address to all the larger offices a Schedule intended to elicit such replies as would enable me to present a general idea of the methods applied to these cases.

In this I hope I have partially succeeded, but I need not add that the information must be regarded in a very general light. Many offices were particular to point out that, in the absence of any ascertained experience, each case of this nature would be considered upon its individual merits, and rated accordingly.

When Mr. Manly first put forward the idea of constructing a Model Office for whole of life policies, endowment assurance business was in its infancy. That idea was afterwards improved on by Mr. King, who substituted an actual experience of entrants and withdrawals for the assumptions which Mr. Manly, in the absence of data, was forced to make; and when, twenty-five years later, another Model Office was constructed on the basis of the new experience, the age distribution of the business was found to be so little altered, that it was possible to retain the older figures. In the meantime, however, endowment assurances had ceased to be a minor class; and their growth has been so rapid and continuous that they now form one of the most important sections of the business. It was probably the hope of many that Mr. King would have found time to extend his Model Office to endowment assurances, and it was only because he stated that his other duties would not permit him at present to do so, that I have undertaken to supply the extension.

In September 1905, at the request of Director Sven Palme, 1 began an investigation into the mortality among Scandinavian emigrants to the Congo, based upon a portion of the Statistical Reports contained in a work entitled “Scandinavians in the Congo”, published in the summer of 1905 by the Danish Lieut.-Colonel H. Jenssen-Tusch. I submit herewith the following account of the manner in which the investigation in question was carried out.

The material is gathered from the “Chronological Record” respecting 922 Scandinavian Congo Emigrants 1878-1904” contained in “Supplement No. 1” to “Scandinavians in the Congo.” This record gives the number, nationality, name, occupation and date of entry for all Scandinavians emigrating to the Congo in each of the years from 1878 to 1904. The records as to date of birth and date of exit from the material for observation have been obtained partly, and to the greater extent, from “Scandinavians in the Congo”, further from the “Swedish Mission Society's” Register of Missionaries sent out, and, finally, from official statements prepared ministerially by the General Secretariat of the Congo State.

The whole material is thus constituted of 922 persons, of whom 236 died during the period of observation—of these, 24 during the journey home from the Congo or later on as a result of residence in the Congo. From this material the whole number of entrants in the year 1904 has been separated-altogether 60 persons—because for these, with the exception of information as to the date of entry, the other necessary data were wanting in nearly every case.

It is now some years since a paper was presented to the Institute dealing with the comparative bonus-earning powers of whole life and endowment assurances. In view of the importance of the subject I have ventured to submit this paper, which I hope may lead to a useful and interesting discussion. The matter has been so ably dealt with by the late Mr. Sunderland, Mr. Lidstone and Mr. Chatham that I have the greatest hesitation in presenting my own views, and I must therefore beg the Members of the Institute to receive this paper in the same kindly spirit which they have always manifested towards new contributors.

My object in the paper has been to show the relative bonuses which should be allotted to whole life and endowment assurance policies having special regard to

(1) The expenses of the office;

(2) Variations in the rate of interest earned; and

(3) Mortality.

In Table A are given the office premiums which have been employed in this investigation. They are based on the H[M] 4 per-cent Table, with a loading for a £1 per-cent compound reversionary bonus, together with 10 per-cent and a constant of 3s. per-cent for expenses and commission—

The main object of my paper on the Error introduced into Mortality Tables by Summation Formulas of Graduation, published at page 54 of the present volume of the Journal, has already been to a large extent fulfilled; and the admirable papers which it has called forth from Messrs. Lidstone and Spencer are a liberal reward for my labours. That object was, to show that the theoretical error in question is so small as to be quite insignificant; and thereby to bring summation formulas up again for discussion, with a view to ascertain which of them are the best as means of graduation pure and simple, and how they can be used most effectively.

In the concluding sentences of the paper I spoke of a graduation by summation, to be followed, in order to gain smoothness, by one by the graphic method; and in the reply to the discussion I casually threw out the idea of correcting the theoretical error by means of the fourth central difference; but these suggestions were only concessions to the misgivings of weaker brethren, and were not put forward seriously as being in my own opinion either necessary or desirable. I am convinced that we have now summation formulas which give as smooth graduations as need be desired, and the results of which conform with close accuracy to the original facts; and, emphasizing what has already been said, my own view is that the theoretical error they introduce is so minute as to be unworthy of consideration for a moment.

The controversy which took place a few years ago (J.I.A. xxvi, 77 and 420 ; xxix, 59, 232 and 236) between Dr. Sprague and the late Mr. Woolhouse on Summation Formulas for the graduation of mortality tables, and more particularly on that of Mr. Woolhouse himself, left the question in a very unsatisfactory state, as a few extracts from the writings of these two great, but divergent, authorities will sufficiently show:

Dr. Sprague, xxvi, 111, said, “We must include in one “general condemnation all such graduation formulas as “Mr. Woolhouse's, Mr. Higham's, and Mr. Ansell's. They all “have a tendency to distort the true law of the facts”; and, again, xxix, 61, “I see no sufficient reason for distinguishing “between his (Mr. Woolhouse's) formula, and others of a like “kind ; and I include them all in one general condemnation.”

In contradiction of this, Mr. Woolhouse, xxvi, 424, said, “The method (his own) stands alone, as systematically based on “rational principles”; and, again, xxix, 241, “I emphatically” repeat that, with practicable data, the application of my system “of adjustment to a mortality table will always give the best ”possible results.”

It has often occurred to me that but scant justice has been done to the application of the infinitesimal calculus to the theories of compound interest and life contingencies. This is, perhaps, in some measure due to the popular relegation of the differential and integral calculus to the realms of the so-called “higher mathematics.” There are, of course, two aspects of the case to be borne in mind. On the one hand, it is necessary to present the subjects in such a form as will be best suited to the student who is commencing to study them. For this purpose experience shows that a start should be made with particular cases, leaving the generalization until such time as the student shall have obtained a grasp of first principles sufficient to enable him to view the subjects in their general aspect. On the other hand, however, there is no doubt that to the reflective mind there comes a time when the desire is felt to invert the process and deduce the formulæ in their logical sequence from a fundamental general hypothesis.

So much has been written on the algebra of formulæ for graduating an irregular series by means of successive summations, and on the theoretical errors introduced by this process, that there is possibly some danger of the underlying principles, on which all such formulæ are founded, being overlooked or insufficiently understood. In the present note an attempt is made to deal with this matter from first principles, with the object of rendering clear how such formulæ Work and why it is that they produce, with greater or less success according to the nature of the particular formula, a smooth succession of values from rough and irregular data.

Mr. George King's recent investigation of the magnitude of the error introduced into certain functions of a mortality table following Makeham's law when these are re-graduated by summation formulas (J.I.A., xli, 54), has again attracted attention to the subject of the graduation of mortality tables, and has recalled the lively controversy which took place some years ago between Dr. Sprague and the late Mr. Woolhouse over the merits and demerits of the methods championed by those distinguished authorities. Not the least interesting contribution to the discussion which followed the reading of Mr. King's paper was furnished by Dr. Sprague himself, who wrote in defence of certain criticisms expressed by him with regard to methods of summation generally and Mr. Woolhouse's formula in particular. Dr. Sprague remarked that he had “objected to Mr. Woolhouse's “formula of graduation,on two grounds, which it has in common “with all other summation formulas; that is to say, (1) because “it does not remove the irregularities of the original series of “facts, but only reduces them, and (2) because it distorts the “law of the facts to a greater or less extent.” He went on to say that the results of his re-graduation of the HM (Text-Book) Table by Woolhouse's formula had led him to the conclusion that the second objection might be disregarded in practice, but that there still remained objection (1), which he considered by far the more important.